Consumer price inflation excluding food and energy often performs worse than other measures of underlying inflation in out-of-sample tests of predicting future inflation or tracking an ex-post measure of underlying trend inflation. Nonetheless, inflation excluding food and energy remains popular
for its simplicity and transparency. Would excluding different items improve performance while maintaining the simplicity and transparency? Unfortunately, probably not. Averaging across a series of tests suggests that knowing what items to exclude before seeing the data is problematic and excluding
food and energy is not a bad ex-ante guess. However, ex-post it is not difficult to construct an index which performs considerably better than excluding food and energy.

JEL Classification: E31

Faced with the high volatility of monthly and quarterly overall consumer price inflation, some economists and policy makers have sought an inflation measure which reduces the transitory volatility while maintaining the signal that changes in inflation can imply for the state of the economy. The
most common of these "core" measures is consumer price inflation excluding food and energy, a type of exclusion index. Exclusion indexes are created by taking an overall price index, dropping a pre-determined subset of items, and creating a new price index with the remaining items. While consumer
prices excluding food and energy is the most common exclusion index, other countries emphasize different exclusion indexes. For example, the core price index at the Bank of Canada excludes indirect taxes and eight of the most volatile components.2 The Bank of Japan excludes only fresh food.

Key features of an exclusion index are that the excluded items are determined before the release of the monthly inflation data and that the list of excluded items does not change often. Further, for the items that remain in the index, there is no down weighting or up weighting of individual
items - their relative weights are the same weights used to construct the overall index. These features make exclusion indexes fairly easy for the general public to understand and for analysts outside the statistical agencies to replicate, and they are important reasons for the popularity of
exclusion indexes as core inflation measures.

However, exclusion indexes are just one of many possible ways of a constructing an index of "core" inflation, a concept coined by Gordon (1975) to describe an underlying inflation rate as opposed to a temporary inflation movement. Further, it is not clear that exclusion indexes are the best
way to construct a core inflation index, or that the consumer prices excluding food and energy is the best exclusion index for the US. Rich and Steindel (2007) suggest that no core inflation measure consistently performs better than other measures at tracking an ex-post measure of trend inflation
or predicting future inflation over time. While other studies have supported this point, they have generally found consumer prices excluding food and energy to perform somewhat worse than other inflation measures. Among studies using US data, Smith (2004) found that weighted median inflation
performs better than consumer prices excluding food and energy at predicting future inflation, both in-sample and out-of-sample. Similarly, Meyer and Pasaogullari (2010) found that either the trimmed mean CPI or inflation expectations from the Survey of Professional Forecasters generally were
better predictors of future CPI inflation than the CPI excluding food and energy. Further, Detmeister (2011) found that PCE inflation excluding food and energy performed worse at matching a handful of ex-post PCE inflation benchmarks than a number of alternative approaches to core inflation.
34

One response to the relatively poor performance of inflation excluding food and energy is to emphasize a different type of measure, such as the trimmed mean, which removes the items with the highest and lowest inflation rates each month, or variance-weighted inflation, which puts less weight on
items which have highly variable inflation. However, given the popularity and simplicity of consumer prices excluding food and energy, this note attempts to improve that index rather than abandon it. The primary method used to tease out an index which better predicts future US headline inflation
and better tracks ex-post trend inflation is to simply sequentially remove items one by one.

The lessons from this attempt to create a better exclusion index of US PCE inflation are:

An optimized in-sample exclusion index would exclude nearly all energy items, but less than half of food consumption. Aside from food and energy, it would also exclude around 2/3 of other goods consumption and 1/3 of other services consumption-dropping items such as used car margins, tobacco,
furniture, life and car insurance, imputed financial services, and most transportation services. This index performs better in-sample than all other core inflation measures to which it is compared.

Out-of-sample, however, an exclusion index optimized based on in-sample fit performs no better than excluding food and energy, or excluding only energy. It also performs far worse than many alternative core inflation measures such as trimmed means and component smoothing. Instability in the
covariance of inflation rates across items suggests an exclusion index approach is unlikely to be a fruitful method of creating a substantially better core inflation measure for out-of-sample use.

I reach these conclusions by examining a combination of 60 different specifications to decide which items to exclude from the index. These specifications differ along three dimensions: 1. the sampling interval used (price changes over 1, 3, 6, or 12-months), 2. the time period examined (December
1978 to September 2009, March 1991 to September 2009, or January 2000 to September 2009), and 3. the ex-post benchmark that the index is design to fit (a centered 36-month moving average of overall inflation, a Baxter-King band-pass filtered version of overall inflation, overall inflation over the
next 12 months, overall inflation 12 to 24 months ahead, or forecasting 12-month ahead overall PCE inflation in regression which includes Phillips curve driving variables).

The items and the share of the consumption bundle excluded are sensitive to the choice of specification. However, the finding that an optimized exclusion index performs little better than standard exclusion indexes in out-of-sample exercises, and worse than some other methods such as trimmed
means and component smoothing, appears robust to a number of alternatives.

The focus here is only on US inflation and specifically on inflation in the personal consumption expenditures (PCE) price index published by the Department of Commerce's Bureau of Economic Analysis (BEA) as part of the National Income and Product Accounts.5 However, since the majority of PCE prices are derived from items in the more-well-known Consumer Price Index (CPI) published by the Department of Commerce's Bureau of Labor Statistics, the main
conclusions of this note are likely to hold for the CPI as well.

Constructing an optimized exclusion index

Previous studies by Clark (2001) and Dolmas (2009) looked at the effects of excluding different item from overall inflation.6 Clark (2001) used a 36-category
breakdown of CPI prices, ranked them by their monthly volatility over the 1967-1997 period, and excluded the eight most volatile categories. He found the resulting core index less appealing than either a trimmed mean or the CPI excluding energy. Dolmas (2009) used a 186-component disaggregation of
PCE prices to exclude 19 items for which their individual exclusions from the headline index lead to a reduction in the cyclical volatility of the inflation index over the 1987 to 2008 period. He found that these exclusions led to an index which more closely tracked an ex-post measure of trend
inflation and better forecasts of future inflation over the 1987 to 2008 period. Dolmas did not try his procedure on an out-of-sample basis, but did find that the list of items excluded in a 1997 to 2008 sample differed only slightly from the full 1987 to 2008 sample.

Similar to these studies, this current note uses a more disaggregated price structure than simply food, energy and other items. Using a disaggregated categorization of PCE can provide a better sorting of the individual components into items that should be included and items that should be
excluded from inflation to make the resulting index best match future or current trend inflation. The 205-item breakdown of PCE prices used in this note-the most disaggregated breakdown available over the time periods examined-is similar to the disaggregation used in Dolmas.7

The studies of Clark and Dolmas determined which items to exclude by focusing on variance. Clark excluded the items with the most volatile monthly inflation rates. The idea is that the high volatility of these items adds noise to the inflation index. Dolmas took this variance approach a step
further by recognizing that the increased variance that an item adds to the inflation index depends on what other items are in the index. He examined changes in the variance of the aggregate inflation index resulting from excluding items one at a time. However, by focusing on reducing volatility
these studies ignore that volatility is not the only important aspect for a core inflation index. Besides volatility (noise) price changes also contain a signal for future inflation, and reducing the variance of the resulting index can also reduce its signal.

To see the signal-noise trade off more clearly, assume that we measure the usefulness of the core inflation index by the standard error of the core index from some benchmark inflation series, perhaps a measure of ex-post trend inflation or top line inflation over the next twelve months.8 This standard error can be written as

(1)

where
is the core inflation rate between time t-s and t, and
is the benchmark inflation rate that the core index is trying to match at time t.

The standard error of equation 1 is minimized by reducing the noise of the core index,
, while maximizing its signal,
. (The variance of the benchmark,
, is the same regardless of the composition of the core index.)

Decomposing these last two terms in equation 1 into individual items in the inflation index we can write9:

(2)

where
is the annualized percent change in price for item i in the index between time t-s and t,
and w is the weight of the item in the core inflation index. The last line of this equation shows that the standard error in equation (1) is minimized by
excluding items which co-vary strongly with other items in the index, but do not co-vary strongly with the inflation benchmark. In other words, items that add more noise (
) than signal (
) to the core index should be excluded.

Note that the basis for the approach used by Clark, and also used to determine the Bank of Canada's core index, is a special case of the result in equation 2. To see this, assume that the noise is uncorrelated across items,
for all items in equation (2), and that the underlying signal contained in each item is the same,
for all items. Then the right-hand side of equation 2 could be re-written as
, which is minimized by removing the items with the highest variance.

The more general approach taken in this note does not assume either a common signal or uncorrelated noise across items. Instead, it directly minimizes equation (1) by sequentially eliminating items. As a result the procedure takes into account both differences in signal and correlations in noise
across items.

The procedure starts with all items in the PCE price index and iteratively removes the item which gives the maximum reduction in the standard error per unit of consumption. That is, at each step all components in the index are ranked by

where the standard error of the core index is defined as in first line of equation (1). The item at the top of this ranking is removed and a new core index is constructed which takes all previous exclusions as given. All items are re-ranked using the new core index as the baseline and the process
is repeated until there are no more exclusions that will reduce the standard error of the index.10 This process is similar to a steepest ascent gradient
method of optimization with the goal of maximizing the fit of the index while minimizing the share of the consumption basket excluded.11

In-sample results for a sequential exclusion price index

Following the procedure outlined above, table 1 lists the first 20 items (out of 205) excluded using four different specifications. These specifications differ in the interval used to determine the inflation rate in the core index (1- or 12-month price changes) and in the benchmark used to
determine the fit of the core index (a centered 36-month moving average of top line inflation and inflation over the next 12 months). All four specifications use the same time period which runs from the beginning of 1978 through September 2009.

Five items are among the first 20 removed in all specifications: Gasoline and other motor fuel, natural gas, fuel oil, pork, and dishes and flatware. Five other items are among the first 20 items excluded in three of the four specifications: intercity buses, household linens, information
processing equipment, public air transportation, and telephone and facsimile equipment. The remainder of the first 20 items removed varies significantly by the specification.

Table 1. First 20 Items Removed Using Four Different Specifications: In-sample over 1979 to 2009 time period

The four specifications shown in table 1 are just a small sampling of possible specifications that could be used and the results suggests that different specifications will lead to different items being excluded, though some items are likely to be excluded in most specifications. To create a
fairly general specification that is not over-optimized for one particular specification this note examines 60 specifications which differ by the time period examined, the interval used to determine the inflation rate in the core index, and the benchmark used to determine the fit of the core
index.

Three different time periods are examined: January 1980 to September 2009, March 1991 to September 2009, and January 2000 to September 2009. These time periods were chosen to align with Detmeister (2011), which will be helpful in comparing the core index created here with alternative measures of
core inflation. The last time period, January 2000 to September 2009, represents the data available since a considerable revision to the CPI methodology in 1999.

Four different sampling intervals are used: 1, 3, 6, and 12-month price changes. These intervals are the most common at which inflation is referenced.12

Five different benchmarks are consulted to judge the fit of the core index: a centered 36-month moving average of overall inflation, a Baxter-King band-pass filtered version of overall PCE inflation, overall PCE inflation over the next 12 months, overall PCE inflation 12 to 24 months ahead, and
forecasting overall PCE inflation in next 12-months using regression which includes Phillips curve driving variables. These benchmarks are meant to align with the main uses of a core inflation index as either an indicator of the current inflation rate purged of transitory factors (centered 36-month
moving average of overall inflation and a Baxter-King band-pass filtered version of overall inflation) or a predictor of future inflation (overall inflation over the next 12 months, overall inflation 12 to 24 months ahead, and forecasting overall PCE inflation over the next 12 months in regression
which includes Phillips curve driving variables13). These tests are more fully described in Detmeister (2011).14

Mixing and matching these choices leads to 60 unique specifications. We can get an idea of how similar to specifications are, and hence what choices are the most important, by examining cross-correlations in the order of when an item was excluded over the specifications. The 60 specifications
result in 1,770 bilateral correlations.

The average rank-order correlation across the specifications is 0.29, suggesting that different specifications lead to quite different exclusion indexes. If two specifications differ by only one dimension the average correlation is 0.46, if they differ across two dimensions the average
correlation drops to 0.30, and if they differ across all three dimensions the average correlation falls to 0.22. Panels A, B, and C of table 2 show the average rank-order correlations for specifications that differ along one dimension.

Table 2. Bilateral Rank-Order Correlations for Specifications which Differ by a Single Dimension

Panel A. Average correlations where specifications differ only by benchmark

Benchmark: Centered 36-month moving average

Benchmark: Inflation in next 12 months

Benchmark: Inflation 12 to 24 months ahead

Benchmark:Phillips curve

Baxter-King band pass filter

.70

.72

.31

.15

Centered 36-month moving average

.66

.51

.18

Inflation in next 12 months

.41

.17

Inflation 12 to 24 months ahead

.14

Panel B. Average correlations where specifications differ only by sampling interval

Length of sampling interval:

3 months

6 months

12 months

1 month

.73

.46

.33

3 months

.68

.48

6 months

.71

Panel C. Average correlations where specifications differ only by time period

Time period starting date:

March 1991

January 2000

January 1980

.49

.31

March 1991

.54

Using either the Baxter-King band pass filter of inflation, a centered 36-month moving average of inflation, or inflation in the next 12 months as the benchmark inflation series seems to give roughly similar results. Rank order cross correlations for specifications which only differ among these
three benchmarks average about 0.7. However, changing the benchmark to either inflation over months 12 to 24 in the future or a Phillips curve lowers the rank order correlation, suggesting that that these benchmarks may lead a considerably different core inflation index than the other
benchmarks.

Similarly, small changes in the sampling interval lead to roughly comparable core inflation indexes, but large changes in the sampling interval can lead to consequently larger changes in the resulting index. For example, the rank order correlation is .7 between specifications which differ only
by the sampling interval being 1 or 3 months, but falls to .3 when the difference in sampling interval is 1 and 12 months. Differences in time periods, panel C, can also lead to quite different core inflation bundles. These results suggest all three specification choices-benchmark, time period, and
sampling interval - are important and can lead to quite different bundles of items to exclude.

The low correlation of the list of items to remove suggests that a stable group of items to exclude may be elusive. It also creates a bit of a dilemma: Given the number of different ways that core inflation is used, there is no obvious way to choose which of these 60 specifications, or numerous
other possible specifications, makes the most sense to use to determine which items to exclude. Therefore, instead of choosing a single specification, the 60 different specifications are averaged. Averaging should result in a price index that is generally applicable across different benchmarks,
time periods, and sampling intervals.

and the sequential exclusion process is re-run using this average criteria to determine which items to remove at each step. The logarithm of the deviations are used when averaging across specifications to keep specifications with high standard errors, such as specifications with a one-month
sampling interval, from dominating which items should be excluded.

The full order of exclusion using this combined specification is shown in the appendix table. Not surprisingly, gasoline, fuel oil, and natural gas are the first three items removed out of 205. Electricity is the twenty-second item removed. These four items comprise more than 98 percent of
consumer energy expenditures and they are all removed in the first 12 percent of the consumption bundle excluded. Their early exclusion suggests that the normal practice of simply excluding the all energy items has some merit.

On the other hand, only some food products are among the earlier items removed. The first food items removed are fats and oils (10), processed dairy products (13), eggs (15), fresh vegetables (18), and pork (24). Many food items are not removed until much later. Sugar and sweets (171), beer
(174), and fish and seafood (183) are among the last 20 percent of the consumption bundle excluded.15 Further, food purchased at restaurants (food in
purchased meals - other purchased meals (205)), is the very last items removed using this sequential exclusion procedure.16

Aside from food and energy, many of the exclusions are consistent with informal views of which items have highly transitory shocks that can swing around PCE prices: Used car margins, imputed banking services, airline fares, and tobacco are among the first 15 percent of the consumption bundle
excluded. Similarly, some items which are informally thought of as being more persistent are among the later items removed, such as owner's equivalent rent (166), tuition for higher education (195), and a number of other services.17 There are, however, some surprises: Lodging away from home (hotels & motels - 141) and new cars (foreign - 170 and domestic - 176) are removed much later than would be suggested by simply looking at the
variability of the items' price changes.

Figure 1 shows how the order of the items exclusion using this average criteria compares to the variance of that item's monthly inflation rate over the March 1991 to September 2009 period, which proxies for the type of criteria used by Clark. As can be seen from this figure there is some
relationship between how quickly an item is excluded and the variance of its one-month inflation rate, but the relationship only holds for the first 40 to 50 items excluded. This suggests that the methodology used here will lead to a somewhat different price index than using only the item's
variance to determine whether it should be excluded or not.

Figure 1: Item Variance Versus the Order of Exclusion

To show how well this new measure fits, figure 2 shows the how the geometric average of the standard errors for the sixty specifications changes as the share of consumption excluded increases. As can be seen from the figure, the average standard errors from the newly-created exclusion index drop
quickly as the first few items are removed and are minimized when just over half of the consumption bundle is excluded (and 128 of the 205 items) - marked by the small vertical line. However, the average change in error is pretty small once a third of the consumption basket has been removed. Simply
removing gasoline or energy items improves the fit of the index a large amount without removing much of the consumption basket. At the point where the average standard error from the sequential exclusion procedure hits its minimum the standard error of the index is about half of the standard error
of the all-item PCE index. Once more than about half of the consumption basket is removed the average standard error increases, but the pattern is not uniform.

Figure 2: In-Sample Fit as Items are Excluded

Table 3 shows the categorization of the items excluded at the minimum point. All energy items are removed, and around three-quarters of food consumption is excluded. Between two-thirds and three-quarters of other goods consumption is removed, but only between one-third and one-half of non-energy
services. Among the goods, non-durable goods are more likely to be removed than durable goods. Among services, housing services are less likely to be removed than other services.

Table 3. Classification of Items Excluded from the In-Sample Version of the Sequential Exclusion Price Index

Category Share of 2000 to 2009 Overall PCE

Share of Category Consumption Excluded

Number of Category Items Excluded/ Category Total

Overall PCE

100

54

128 / 205

Energy

5

100

6 / 6

Food and Beverages

7

75

15/22

Food

6

83

14 / 19

Alcohol

1

26

1 / 3

Excluding Food and Energy

87

49

107/177

Goods

24

72

42 / 67

Durable Goods

12

68

25 / 40

Motor Vehicles

4

49

4 / 9

Other Durable Goods

8

79

21 / 31

Nondurable Goods

12

77

17 / 27

Apparel

4

82

3 / 6

Other Nondurables ex. Food and Energy

8

75

14 / 21

Services ex. Food and Energy

63

40

65 / 110

Housing

16

24

6 / 9

Medical Care Services

15

50

4 / 8

Other Services

33

44

55 / 93

Market-based

20

47

29 / 50

Non-market

13

39

26 / 43

Other non-market services includes categories which have some market-based and not-market-based service components

In-sample results also suggest that the sequential exclusion procedure also performs better than a number of alternative measures of underlying PCE inflation, which are also shown on figure 2.18 At its minimum point the error from the sequential exclusion procedure is below the error of all of the other measures. The majority of these other measures do not use an exclusion-based framework, and as noted at the beginning
of this paper, the exclusion based measures (excluding gasoline, excluding energy, excluding food and energy, sticky price items, services only, and Dolmas' exclusion) generally perform worse than many of the other measures. However, Dolmas' exclusion performs notably better than the other
exclusion indexes and, arguably, begins to approach the performance of the trim-based measures (Dallas Fed trimmed mean, average trimmed mean, trimmed mean of volatile components, weighted median), though it still does not perform as well as the sequential exclusion procedure.19

Much like the results in Dolmas (2009), these results show that once all the data are known we can construct an exclusion index of core inflation that is much better than simply excluding food and energy. In fact, once the data is known we can create an exclusion index of core inflation that
performs better than any of the other any of the other main indexes of underlying inflation. A much harder test is to see whether we can create a better core index before observing all of the data.

Out-of-sample results for a sequential exclusion price index

Transforming the in-sample sequential exclusion index used above into an out-of-sample version requires some modification of how to determine the basket of items excluded. The primary choice is whether to hold the items excluded fixed or allow them change over time. A clearly defined and stable
basket of items is one of the features that make an exclusion index transparent and popular. However, since the economy's structure is continuously changing it likely makes sense to allow the basket of excluded items to evolve over time. Trading off these two competing ideas suggests allowing the
basket of items to change only occasionally. As a result when creating the out-of-sample version of the sequential exclusion index the basket of excluded items is allowed to update once every five years.

Specifically, the out-of-sample sequential exclusion index is constructed by determining the best-fitting index using data through December 1978 and then using this basket of items for January 1979 to December 1983. The basket is then updated using data through December 1983 and the new basket
is used from January 1984 through December 1988. The process is repeated every five years. The same five benchmarks and four sampling intervals from the previous section are used here to construct the out-of-sample index, but the three time periods are altered to the most recent 8, 15, and 30 years
(or all available data if less than 30 years is available).

Following this out-of-sample procedure the average standard error across the sixty specifications of the sequential exclusion rises to 0.80 percentage point on annualized PCE inflation. This 0.80 percentage point out-of-sample error compares poorly to the 0.62 percentage point obtained in the
in-sample procedure and shown in figure 1. Also, this out-of-sample standard error is nearly identical to the standard error for excluding food and energy, shown in figure 1, and it is far worse than the best-fitting alternative index, the Dallas Fed trimmed mean which has an average standard error
of 0.67 percentage point.20 Thus strong results obtained in-sample do not carry over out-of-sample and out-of-sample the sequential exclusion index performs
no better than standard core inflation measures.

A number of variations on the procedure were examined in order to attempt to improve the out-of-sample results. These variants included: Excluding only the first 5, 10, 15, 20, or 25 percent of the consumption basket (rather than around 50 percent of the consumption basket that is excluded when
maximizing fit in-sample); Averaging errors over different years when determining exclusions to reduce the influence from one or two outlier; Determining exclusions in a non-path dependent (i.e. non-sequential) manner, that is by looking at the effect of removing single items from the overall price
index rather than the effect of removing the items from an index which has had previous exclusions; Altering the time periods used in determining which items to exclude; and increasing the frequency of updating the basket of items from once every five years to once year or once every 3 years. While
some of these alternatives did slightly improve the fit of the resulting exclusion index, none resulted in an index which fitted more than small handful of basis points better.

The poor out-of-sample performance of exclusion indexes is not confined to the sequential exclusion methodology that has been the focus of this note. Clark did not find that excluding items based on their volatility led to out-of-sample performance that improved upon simply excluding energy.
Similarly, an out-of-sample version of Dolmas' procedure also performs about the same as excluding food and energy.21

The fragility of out-of-sample exclusion indexes

Why do out-of-sample exclusion indexes perform poorly? In short, for most items the historical variances and covariances used in determining whether the item should be included or excluded are only moderately helpful for predicting those covariances in future periods.

For the sequential exclusion index that has been the focus of this note, equation 2 above shows that the determination of which item to exclude depends on the item's covariance with the current price index and its covariance with the benchmark used to proxy for underlying inflation. Strong
out-of-sample results require these covariances to be relatively predictable, but in practice they are not.

To illustrate this point, figure 3 displays the covariances of an item's price change with total PCE inflation over the past 10 years (the horizontal axis) compared to the covariance over the following five years (the vertical axis). The top panel uses a one-month sampling interval, while the
bottom panel uses a 12-month interval. These panels serve as a check of the noise component in equation 2. At either sampling interval there is a positive relationship, but the relationship is not terribly strong. At the one-month interval the slope coefficient is around two-thirds with a standard
error of about 0.05 and an R-squared of 0.12. At the twelve-month interval the slope coefficient is much smaller at 0.2, with a standard error of 0.02, and an R-squared of 0.07. These results suggest that there is some predictive power from an item's past relationship with total inflation for the
item's future relationship with total inflation. However, past covariances accurately predict only a small part of future covariances.

The item's signal is even less stable. The covariance between the change in the price of an item and underlying inflation, measured here as a Baxter-King band pass filter of headline inflation, bears little relationship to that covariance in the next period. Figure 4 shows these covariances
using the one-month sampling interval for the core index in the top panel and the 12-month sampling interval in the lower panel. Using one-month inflation rates the R-squared is .14, while it is a fairly dismal .01 using twelve month inflation rates. This suggests that the signal that an item has
for future benchmark inflation is not closely correlated with the signal that the item had over the prior ten years.

These findings do suggest that past covariances are some help in predicting future covariances, thus exclusion indexes are likely to be of some use as a core inflation measure. However, the past covariances are not a strong proxy for future covariances. This relative instability suggests that
the amount of improvement we would expect to see from an exclusion approach to core inflation is limited, and as a result alternative methods probably hold more promise for improving core inflation measurement.

Conclusion

Previous studies of core inflation measures suggest that no single core inflation measure consistently performs better than others, but inflation excluding food and energy has often been found to perform worse than other approaches to core inflation. Nonetheless, inflation excluding food and
energy remains the most popular core inflation measure in the United States. This study attempted to tweak the basket of excluded items in order to improve the performance of the resulting index at predicting future inflation and at tracking an ex-post measure of trend inflation.

Changing the items excluded can lead to a better in-sample core index, with the items excluded similar to the items often judgmentally discounted because of their high variance. Nevertheless, the approach used to decide what items to exclude ultimately fails out-of-sample: The resulting index
still performed no better than excluding food and energy. The reason for the failure appears to be that past covariances of individual items either with other items in the core index or with an inflation benchmark are only modestly helpful at predicting future covariances.

The instability of these covariances suggests that relying solely on an exclusion index approach is unlikely to yield significant improvements in the measurement of core inflation. Other approaches, such as the trimmed mean, may provide a better single measure of non-transitory inflation.

* The views expressed herein are those of the author and do not necessarily reflect those of the Federal Reserve Board or its staff. For helpful comments and assistance I thank Monica Savukinas, Deb Lindner, David Lebow, Dan Sichel, Peter Tulip, John Roberts,
Michael Kiley, Julie Smith, Jim Dolmas, Todd Clark, Joseph Chien and numerous Federal Reserve Board staff. Return to Text

3. The alternative approaches to PCE inflation examined by Detmeister (2011) included various exclusion indexes, trimmed mean and weighted medians, variance-weighting inflation, weights based on regression coefficients, cost of nominal distortions weighting
(CONDI), trend inflation from Stock and Watson's UC-SV model, Michigan inflation expectations, and component smoothing. These alternatives will be compared to the current approach later in this paper, most notably in figure 2. Return to Text

5. The choice of using the PCE price index, rather than the CPI, was driven by PCE item structure. When the BEA revises the PCE item structure the revision of categories is carried throughout the history of the index. In the CPI, such category changes are only
done on a forward going basis. A uniform disaggregation of sub-headline price categories throughout the history of the price index greatly simplifies the analysis of which items to include or exclude. Return to Text

6. A third paper, Pedersen (2009), used a technique that more closely follows the standard trimming procedures by allowing the items excluded to vary each month and by determining the items to exclude in the month only after observing that month's inflation
data. Return to Text

7. This 205-item breakdown, derived from BEA table 2.4.4U with data available as of mid-October 2011, is the most disaggregated breakdown of PCE prices possible from publicly available data in which every category contains monthly observations back to 1959.
When a component in table 2.4.4U did not have data back to 1959, the lowest aggregate with data back to 1959 was used instead of the individual component indexes comprising the aggregate. A handful of items enter PCE with a negative weight. These items are treated the same as any other except that
their weights are negative. This differs from Dolmas who removes components with negative weights before creating the core price index. Tests using an older vintage of data suggest that the treatment of these items has little effect on the resulting index. Return to
Text

8. Note that using standard error to measure fit allows for a constant inflation differential between the core index and the benchmark series. Return to Text

9. This decomposition assumes the core inflation rate can be written as the weighted average of the component inflation rates. This is true for the Lasperyes index used in constructing the CPI, but it is not exactly true for the Fisher index formula which the
BEA uses to construct PCE prices. Nonetheless, the approximation is close enough to a true Fisher index for the purposes here. Return to Text

10. The approach of computing the standard error before and after removing the item is somewhat more general than removing the item with the highest
at each step as it will account for second order effects of removing an item, such as the
effect that removing an item has on the remaining items' covariance with the new core index. Return to Text

11. Components that are a large share of expenditures will have a large (positive or negative) impact on the variance of the index. If the size of the component was not taken into consideration in equation 3 then items with large weight would tend toward the
beginning or end of the list of items removed, and the subsequent list of items to remove would be very sensitive to the level of disaggregation used. When using formula 3, which does divide by the item's share, there does not appear to be substantial correlation between the size of the item and
its order of removal. As with any gradient method, this approach is path dependent and will find the minimum subject to previous choices. Finding the globally best-fitting index would require re-optimizing all of the components at each step and involve exponentially more computations. Return to Text

12. As shown in Bryan and Meyer (2011) and Detmeister (2011) sampling intervals on the order of 12 months or more usually track ex-post in more closely and predict future inflation better. However, given core inflation is often referred to at shorter sampling
intervals we include these in the analysis as well. We average across all four sampling intervals to avoid having different inflation baskets for different sampling intervals, which could cause confusion and would hamper the index's usefulness for communication. Return to Text

13. The variant of the Phillips curve in this exercise is
where the driving variables are the unemployment rate gap and non-fuel imports and oil price changes weighted by their share in personal consumption expenditures, as well as their lags of these prices,
. Note that this form requires estimates of the unemployment rate, oil and import prices over the next twelve months. In the regressions here actual values were used. To match the fit measure of the other benchmarks, the standard error of the estimate is used as the measure of fit for the
Phillips curve. To make sure the core index created in this regression comprises the basket of items that total inflation converges to (rather than moves away from), the coefficient on core inflation is constrained to be non-negative. Return to Text

14. The procedure here differs slightly from Detmeister (2011) in that there is no adjustment for bias. Instead, the standard error setup here does not penalize for a constant bias over the time period. That paper noted that the treatment of bias had little
effect on the results. Return to Text

15. Using the standard definitions of "core" prices, beer, along with all beverages, is excluded from the core PCE price index, but it is not excluded from the core CPI. Return to Text

16. The comprehensive NIPA revision in mid-2009 by the BEA moved food purchased at restaurants out of goods and into services. As a result the category is no longer excluded from the core PCE price index. However, because of differences in organization it
continues to be excluded from the core CPI. Return to Text

17. One reason that some services tend to be more stable might be that the BEA estimates price changes for some items using the overall CPI or the CPI excluding food and energy weighted with an earnings measure. Return to
Text

18. For more on the additional measures in this figure see Detmeister (2011), particularly the appendix to that note. The errors for the additional measures shown here differ from Detmeister (2011) in that here errors are an average of four different sampling
intervals (1-month, 3-month, 6-month, and 12-month). Figures in Detmeister (2011) show a range of sampling intervals from 1 to 18 months. Consumption shares are not applicable for the Michigan next 12-months inflation expectations, variance-weighted, average trimmed mean, or cost of nominal
distortions index. Therefore these measures have been placed at the midpoint of the horizontal axis of figure 2. Return to Text

19. Dolmas' procedure excludes components whose individual exclusion from overall PCE inflation reduces the standard deviation of high frequency inflation, where high-frequency inflation is derived from a Christiano-Fitzgerald band pass filter with
frequencies from 1 to 36 months. The revision to the PCE item structure in 2009 forced some changes to the 19 items which Dolmas excluded, and his procedure has been replicated for this study using the 205-component breakdown that is used in the sequential exclusion procedure. The different item
structure and a somewhat different time period results in 31 exclusions. The low net share of consumption excluded observed in figure 2, around 3 percent, occurs because 12 of the 31 components excluded are items which have a negative weight in overall PCE inflation. (By far the largest of these
items is nonprofits hospitals' services to households). In absolute terms 18 percent of the consumption basket is excluded. Dolmas' exclusion was not included in Detmeister (2011). Return to Text

20. Some argument can be made that the results for the Dallas Fed trimmed mean may be biased downwards because the trim points were determined using data and tests that are similar to those used here. However, the average trimmed mean, which averages over all
possible symmetric trimmed and therefore does not require trim points to be specified, performs only slightly worse than the Dallas Fed trimmed mean and still far better than any of the exclusion indexes out of sample. Return to Text

21. Like the sequential exclusion procedure, the out-of-sample version of Dolmas' exclusion index was updated once every five years. It has an average standard error of 0.78 percentage points, compared to 0.80 percentage points for excluding food and energy.
Increasing the frequency of updating the index to once a year improved the fit to 0.73 percentage points - the best of the out-of-sample exclusion methods tested, but still worse the 0.67 standard error of the Dallas Fed trimmed mean index. Return to Text